Suppose we wish to test H0: the X and Y distributions are identical versus Ha: the...

Suppose we wish to test

H₀: the X and Y distributions are identical

versus

H_a: the X distribution is less spread out than the Y distribution The accompanying figure pictures X and Y distributions for which H_a is true. The Wilcoxon rank-sum test is not appropriate in this situation because when H_a is true as pictured, the Y’s will tend to be at the extreme ends of the combined sample (resulting in small and large Y ranks), so the sum of X ranks will result in a Wvalue that is neither large nor small.

Consider modifying the procedure for assigning ranks as follows: After the combined sample of m+ n observations is ordered, the smallest observation is given rank 1, the largest observation is given rank 2, the second smallest is given rank 3, the second largest is given rank 4, and so on. Then if H_a is true as pictured, the X values will tend to be in the middle of the sample and thus receive large ranks. Let W′ denote the sum of the X ranks and consider rejecting H₀ in favor of H_a when W′ ≥ c . When H₀ is true, every possible set of X ranks has the same probability, so W′ has the same distribution as does W when H0 is true. Thus c can be chosen from Appendix Table A.14 to yield a level a test. The accompanying data refers to medial muscle thickness for arterioles from the lungs of children who died from sudden infant death syndrome (x’s) and a control group of children (y’s). Carry out the test of H₀ versus H_a at level .05.

Consult the Lehmann book (in the chapter bibliography) for more information on this test, called the Siegel-Tukey test.